Zaman Skalaları Üzerinde Parametreye Bağlı Dirac Sistemleri

Yıl 2018, Cilt: 39 Sayı: 4, 864 - 870, 24.12.2018

Ahmet Sinan Özkan

https://doi.org/10.17776/csj.471958

Cited By: 4

Öz

Bu çalışmada bir zaman skalası üzerinde iki farklı genelleştirilmiş
Dirac sistemi ve parametreye bağlı sınır koşulları ile üretilen bir sınır değer
problemi ele alınmıştır.  Sistemlerin
eşleniksiz (disconjugate) olması için yeterli koşullar ve problemin özdeğerlerinin
sayısı ile ilgili bir formül elde edilmiştir.

Anahtar Kelimeler

Zaman Skalası, Dinamik denklemler, Dirac sistemi, parametreye bağlı sınır koşulları, eşleniksizlik

Parameter-Dependent Dirac Systems on Time Scales

Öz

In this study, we consider two generalized Dirac systems on a time scale
and a boundary-value problem with boundary conditions depending on the spectral
parameter. We give some sufficient conditions for disconjugacy of the systems
and obtain a formula about the number of eigenvalues of the problem.

Anahtar Kelimeler

Time scale, Dynamic Equations, Dirac-System, parameter-dependent boundary conditions, disconjugacy

Kaynakça

• [1]. Hilger, S., Analysis on measure chains – a unified approach to continuous and discrete calculus. Results in Math., 18 (1990) 18-56.
• [2]. Erbe, L., Hilger, S., Sturmian theory on measure chains, Differ. Equ. Dyn. Syst., 1 (1993) 223-244.
• [3]. Agarwal, R.P., Bohner, M., Wong, P.J.Y., Sturm-Liouville eigenvalue problems on time scales, Appl. Math. Comput., 99 (1999) 153-166.
• [4]. Allahverdiev, B.P., Eryilmaz, A., Tuna, H., Dissipative Sturm-Liouville operators with a spectral parameter in the boundary condition on bounded time scales, Electronic Journal of Differential Equations, 95 (2017) 1–13.
• [5]. Amster, P., De Napoli, P., Pinasco, J.P., Eigenvalue distribution of second-order dynamic equations on time scales considered as fractals, J. Math. Anal. Appl., 343 (2008) 573-584.
• [6]. Amster, P., De Napoli, P., Pinasco, J.P., Detailed asymptotic of eigenvalues on time scales, J. Differ. Equ. Appl., 15 (2009) 225-231.
• [7]. Davidson, F.A., Rynne, B.P., Global bifurcation on time scales, J. Math. Anal. Appl., 267 (2002) 345-360.
• [8]. Davidson, F.A., Rynne, B.P., Self-adjoint boundary value problems on time scales, Electron. J. Differ. Equ., 175 (2007) 1-10.
• [9]. Davidson, F.A., Rynne, B.P., Eigenfunction expansions in L2 spaces for boundary value problems on time-scales, J. Math. Anal. Appl., 335 (2007) 1038-1051.
• [10]. Erbe, L., Peterson, A., Eigenvalue conditions and positive solutions, J. Differ. Equ. Appl., 6 (2000) 165-191.
• [11]. Guseinov, G.S., Eigenfunction expansions for a Sturm-Liouville problem on time scales, Int. J. Differ. Equ., 2 (2007) 93-104.
• [12]. Guseinov, G.S., An expansion theorem for a Sturm-Liouville operator on semi-unbounded time scales, Adv. Dyn. Syst. Appl., 3 (2008) 147-160.
• [13]. Hilscher, R.S., Zemanek, P., Weyl-Titchmarsh theory for time scale symplectic systems on half line, Abstr. Appl. Anal., Art. ID 738520 (2011) 41 pp.
• [14]. Huseynov, A., Limit point and limit circle cases for dynamic equations on time scales, Hacet. J. Math. Stat., 39 (2010) 379-392.
• [15]. Huseynov, A., Bairamov, E., On expansions in eigenfunctions for second order dynamic equations on time scales, Nonlinear Dyn. Syst. Theory, 9 (2009) 7-88.
• [16]. Kong, Q., Sturm-Liouville problems on time scales with separated boundary conditions, Results Math., 52 (2008) 111-121.
• [17]. Rynne, B.P., L2 spaces and boundary value problems on time-scales, J. Math. Anal. Appl., 328 (2007) 1217-1236.
• [18]. Sun, S., Bohner, M., Chen,S., Weyl-Titchmarsh theory for Hamiltonian dynamic systems. Abstr. Appl. Anal. Art., ID 514760 (2010) 18 pp.
• [19]. Bohner, M., Peterson, A., Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2001.
• [20]. Bohner, M., Peterson, A., Advances in Dynamic Equations on Time Scales, Birkhauser, Boston, MA, 2003.
• [21]. Gulsen, T, Yilmaz, E, Spectral theory of Dirac system on time scales, Appl. Anal., 96-16 (2017) 2684-2694.
• [22]. Gulsen, T, Yilmaz, E, Goktas, S, Conformable fractional Dirac system on time scales, J. Inequal. Appl, 161-1 (2017) 1-10.